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Mirrors > Home > MPE Home > Th. List > axtgcgrid | Structured version Visualization version Unicode version |
Description: Axiom of identity of congruence, Axiom A3 of [Schwabhauser] p. 10. (Contributed by Thierry Arnoux, 14-Mar-2019.) |
Ref | Expression |
---|---|
axtrkg.p | |
axtrkg.d | |
axtrkg.i | Itv |
axtrkg.g | TarskiG |
axtgcgrid.1 | |
axtgcgrid.2 | |
axtgcgrid.3 | |
axtgcgrid.4 |
Ref | Expression |
---|---|
axtgcgrid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-trkg 25352 | . . . . 5 TarskiG TarskiGC TarskiGB TarskiGCB Itv LineG | |
2 | inss1 3833 | . . . . . 6 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC TarskiGB | |
3 | inss1 3833 | . . . . . 6 TarskiGC TarskiGB TarskiGC | |
4 | 2, 3 | sstri 3612 | . . . . 5 TarskiGC TarskiGB TarskiGCB Itv LineG TarskiGC |
5 | 1, 4 | eqsstri 3635 | . . . 4 TarskiG TarskiGC |
6 | axtrkg.g | . . . 4 TarskiG | |
7 | 5, 6 | sseldi 3601 | . . 3 TarskiGC |
8 | axtrkg.p | . . . . . 6 | |
9 | axtrkg.d | . . . . . 6 | |
10 | axtrkg.i | . . . . . 6 Itv | |
11 | 8, 9, 10 | istrkgc 25353 | . . . . 5 TarskiGC |
12 | 11 | simprbi 480 | . . . 4 TarskiGC |
13 | 12 | simprd 479 | . . 3 TarskiGC |
14 | 7, 13 | syl 17 | . 2 |
15 | axtgcgrid.4 | . 2 | |
16 | axtgcgrid.1 | . . 3 | |
17 | axtgcgrid.2 | . . 3 | |
18 | axtgcgrid.3 | . . 3 | |
19 | oveq1 6657 | . . . . . 6 | |
20 | 19 | eqeq1d 2624 | . . . . 5 |
21 | eqeq1 2626 | . . . . 5 | |
22 | 20, 21 | imbi12d 334 | . . . 4 |
23 | oveq2 6658 | . . . . . 6 | |
24 | 23 | eqeq1d 2624 | . . . . 5 |
25 | eqeq2 2633 | . . . . 5 | |
26 | 24, 25 | imbi12d 334 | . . . 4 |
27 | id 22 | . . . . . . 7 | |
28 | 27, 27 | oveq12d 6668 | . . . . . 6 |
29 | 28 | eqeq2d 2632 | . . . . 5 |
30 | 29 | imbi1d 331 | . . . 4 |
31 | 22, 26, 30 | rspc3v 3325 | . . 3 |
32 | 16, 17, 18, 31 | syl3anc 1326 | . 2 |
33 | 14, 15, 32 | mp2d 49 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3o 1036 wceq 1483 wcel 1990 cab 2608 wral 2912 crab 2916 cvv 3200 wsbc 3435 cdif 3571 cin 3573 csn 4177 cfv 5888 (class class class)co 6650 cmpt2 6652 cbs 15857 cds 15950 TarskiGcstrkg 25329 TarskiGCcstrkgc 25330 TarskiGBcstrkgb 25331 TarskiGCBcstrkgcb 25332 Itvcitv 25335 LineGclng 25336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-trkgc 25347 df-trkg 25352 |
This theorem is referenced by: tgcgreqb 25376 tgcgrtriv 25379 tgsegconeq 25381 tgbtwntriv2 25382 tgbtwndiff 25401 tgifscgr 25403 tgbtwnxfr 25425 lnid 25465 tgbtwnconn1lem2 25468 tgbtwnconn1lem3 25469 legtri3 25485 legeq 25488 legbtwn 25489 mirreu3 25549 colmid 25583 krippenlem 25585 lmiisolem 25688 hypcgrlem1 25691 hypcgrlem2 25692 f1otrg 25751 |
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